Phase Transform for Object and Shape Detection in Digital Images

ABSTRACT

Object and shape detection in digital images utilizing edge detection is described. In a first edge detection approach, phase transformation is utilized in the frequency domain, such as in response to Fourier transform, followed by use of a frequency-domain phase kernel and inverse-Fourier transform. Edge detection is also provided using a phase transform in the spatial domain utilizing a convolution approach. In a second edge detection approach, phase stretching is utilized, such as in combination with phase histogramming along with thresholding and morphological operations. Numerous example images are provided illustrating benefits of the disclosed technology with different applications and under different conditions.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a 35 U.S.C. § 111(a) continuation of PCTinternational application number PCT/US2015/029319 filed on May 5, 2015,incorporated herein by reference in its entirety, which claims priorityto, and the benefit of, U.S. provisional patent application Ser. No.61/988,501 filed on May 5, 2014, incorporated herein by reference in itsentirety, and which claims priority to, and the benefit of, U.S.provisional patent application Ser. No. 62/014,262 filed on Jun. 19,2014, incorporated herein by reference in its entirety. Priority isclaimed to each of the foregoing applications.

The above-referenced PCT international application was published as PCTInternational Publication No. WO 2015/171661 on Nov. 12, 2015, whichpublication is incorporated herein by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not Applicable

INCORPORATION-BY-REFERENCE OF COMPUTER PROGRAM APPENDIX

Not Applicable

NOTICE OF MATERIAL SUBJECT TO COPYRIGHT PROTECTION

A portion of the material in this patent document is subject tocopyright protection under the copyright laws of the United States andof other countries. The owner of the copyright rights has no objectionto the facsimile reproduction by anyone of the patent document or thepatent disclosure, as it appears in the United States Patent andTrademark Office publicly available file or records, but otherwisereserves all copyright rights whatsoever. The copyright owner does nothereby waive any of its rights to have this patent document maintainedin secrecy, including without limitation its rights pursuant to 37C.F.R. § 1.14.

BACKGROUND 1. Technological Field

This disclosure pertains generally to image processing, and moreparticularly to image processing utilizing a phase transform with anoutput phase image for edge detection.

2. Background Discussion

Exponential growth in the amount of digital data generated by sensorsand computers has resulted in a technological problem called “Big Data”bottleneck. One of the most problematic issues when working with BigData is to analyze and make sense out of the huge amount of the floodingdata. In past decades, many computer vision methods, such as edgedetection, object recognition and machine learning algorithms have beendeveloped for Big Data handling.

Edge detection is the name for a set of mathematical methods foridentifying patterns in a digital image where brightness or colorchanges abruptly. Applying an edge detection process to an image is thebasis for numerous forms of object detection, shape, classification,movement detection, and so forth. Edge detection also reduces thedigital file size while preserving important information, albeit datacompression is not the main objective in edge detection.

There are many methods for edge detection, but most of them can begrouped into two categories, search-based and zero-crossing based. Thesearch-based methods detect edges by first computing a measure of edgestrength, usually a first-order derivative, and then searching for localdirectional maxima of the gradient magnitude. The zero-crossing basedmethods search for zero crossings in a second-order derivative computedfrom the image.

Sobel operator is one of the earliest advanced methods developed foredge detection. It is a discrete differentiation operator performed ateach point in the image, the result of the Sobel operator is either thecorresponding gradient vector or the norm of this vector. The gradientapproximation that is produced is relatively crude, in particular forhigh frequency variations in the image.

Other edge detection methods, such as Canny, Prewitt, Roberts, Log andZero cross exist for computer vision applications. The Canny edgedetector, considered as state-of-the-art, uses a multi-stage algorithmto detect edges in an image. Canny uses the calculus of variationstoward optimizing a given function. The optimal function is described bythe sum of four exponential terms, however, it can be approximated bythe first derivative of a Gaussian.

However, even the most advanced Canny edge detection approach suffersfrom a number of shortcomings that limit its ability for discerningedges and objects under the best conditions, and whose results degradesignificantly under adverse image situations and conditions.

Accordingly, a need exists for new edge detection apparatus and methodswhich provide enhanced edge detection abilities which can be utilized inwide range of conditions. The present disclosure fulfill those needs andothers, while overcoming other shortcomings of existing methods.

BRIEF SUMMARY

Two different forms of edge detection in an image(s) under analysis aredisclosed. In a first approach, the image under analysis is passedthrough a Phase Transformation and the output phase image is optionallyPost-Processed to generate an image of the edges. Phase Transformationcan be performed according to the disclosure by operating either in thefrequency domain or spatial domains. Image results are included whichillustrate that the disclosed method is an improvement over Canny edgedetection method in terms of edge detection performance and simplicity.

In a second edge detection method a new computational approach to edgedetection is described, which in some ways emulates propagation of lightthrough a physical medium with specific diffractive property and usesthe resulting phase to identify edges in a digital image. This method isnot iterative and does not require prior knowledge about the image, anddescribes an edge detection process which is more general, including itsuse of pre-processing and use of a localization kernel in detectingedges.

Further aspects of the presented technology will be brought out in thefollowing portions of the specification, wherein the detaileddescription is for the purpose of fully disclosing preferred embodimentsof the technology without placing limitations thereon.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)

The disclosed technology will be more fully understood by reference tothe following drawings which are for illustrative purposes only:

FIG. 1 is a block diagram of image processing utilizing the disclosedphase transform in combination with optional post processing for objectand shape detection according to an embodiment of the presentdisclosure.

FIG. 2 is a block diagram of performing a phase transform in thefrequency domain according to an embodiment of the present disclosure.

FIG. 3 is a block diagram of performing a phase transform in the spatialdomain according to an embodiment of the present disclosure.

FIG. 4 is a block diagram of post processing steps according to anembodiment of the present disclosure.

FIG. 5 is a block diagram of performing object and shape detectionutilizing a phase transform in the frequency domain according to anembodiment of the present disclosure.

FIG. 6A through FIG. 6E are plots of phase kernel and phase derivativerelationships utilized according to an embodiment of the presentdisclosure.

FIG. 7A through FIG. 7D are images (Lena) depicting comparisons betweenan original image using two different Canny methods, as compared withedge detection according to an embodiment of the present disclosure.

FIG. 8A through FIG. 8D are images (Lena hat portion) depictingcomparisons between an original image using two different Canny methods,as compared with edge detection according to an embodiment of thepresent disclosure.

FIG. 9A through FIG. 9D are images (ocean liner) depicting comparisonsbetween an original image using two different Canny methods, as comparedwith edge detection according to an embodiment of the presentdisclosure.

FIG. 10A through FIG. 10D are images (tissue sample) depictingcomparisons between an original image using two different Canny methods,as compared with edge detection according to an embodiment of thepresent disclosure.

FIG. 11A through FIG. 11D are images (brain neural tree) depictingcomparisons between an original image using two different Canny methods,as compared with edge detection according to an embodiment of thepresent disclosure.

FIG. 12A through FIG. 12D are images (child image) depicting comparisonsbetween an original image containing impairment using two differentCanny methods, as compared with edge detection according to anembodiment of the present disclosure.

FIG. 13A through FIG. 13C are images (street in fog) depictingcomparisons between an original image and a Canny method as comparedwith edge detection according to an embodiment of the presentdisclosure.

FIG. 14 through FIG. 17 are images (tissues) showing an original image(FIG. 14) and different parameter selections utilizing an interactiveedge detection for generating different edge enhanced images (FIG. 16through FIG. 17).

FIG. 18 through FIG. 21 are images (woman seated) of a photograph/videoframe showing an original image (FIG. 18) and different parameterselections utilizing an interactive edge detection for generatingdifferent edge enhanced images (FIG. 19 through FIG. 21).

FIG. 22 and FIG. 23 are images (city street) of a photograph/video frameshowing an original image (FIG. 22) and interactive edge detection forgenerating different edge enhanced images (FIG. 23).

FIG. 24 through FIG. 26 are images (planetary body) of an astronomicimage showing an original image (FIG. 24) and different parameterselections utilizing an interactive edge detection for generatingdifferent edge enhanced images (FIG. 25 through FIG. 26).

FIG. 27 and FIG. 28 are images (brain) of a photograph/video frameshowing an original image (FIG. 27) and interactive edge detection ofoverlaid edge data from a medical imaging source in an output image(FIG. 28).

FIG. 29A through FIG. 29D are images (brain MRI) depicting anapplication for utilizing edge detection according to an embodiment ofthe present disclosure, for processing an image or sequence (i.e.,frames of video) with user selected parameters (FIG. 29D).

FIG. 30A through FIG. 30D are images (3D brain MRI) depicting anapplication for utilizing edge detection according to an embodiment ofthe present disclosure, for processing an image or sequence (i.e.,frames of video) with user selected parameters (FIG. 30D).

FIG. 31 is a block diagram of performing object and shape detectionutilizing a phase stretch transform (PST) according to an embodiment ofthe present disclosure.

FIG. 32A through FIG. 32D are images (Lena) depicting comparisonsbetween the Canny method compared and edge detection according to anembodiment of the present disclosure, shown with two differentthresholds and sigma values.

FIG. 33A through FIG. 33D are images (coins) depicting comparisonsbetween the Canny method compared and edge detection according to anembodiment of the present disclosure, shown with two differentthresholds and sigma value.

FIG. 34A through FIG. 34D are images (rocky beach) depicting a normal(un-zoomed condition) and images in response to zooming with edgedetections performed according to an embodiment of the presentdisclosure.

DETAILED DESCRIPTION

1. Phase Transforms for Image Edge Detection

In the disclosed method for edge detection, the image under analysis ispassed through a Phase Transformation and the output phase image isPost-Processed to generate an image of the edges. Phase Transformationcan be performed according to the disclosure by operating either in thefrequency domain or spatial domains. The Phase Transformation in thefrequency domain can be described as follows:

[n,m]=Angle

(IFT{{tilde over (K)}[p,q]·FT{B[n,m]}

  (1)

where B[n m] is the image under analysis,

[n,m] is the output phase image, n and m are two dimensional spatialvariables, Angle< > is the angle operator, FT is the Fourier transform,IFT is the inverse Fourier transform and p and q are two dimensionalfrequency variables. It should be appreciated that other transformswhich can decompose the frequency components of a source signal/imagemay be utilized in place of the Fourier operation, including but notlimited to: Fast Fourier Transform (FFT), Discrete Fourier Transform(DFT), Discrete Sine Transform and Discrete Cosine transforms. The PhaseKernel {tilde over (K)}[p,q] is described by a nonlinear phaseoperation:

K[p,q]=e ^(j·φ[p,q]).  (2)

In general, the Phase Kernel {tilde over (K)}[p,q] can have a number ofshapes, or phase profiles. The general requirement is that the phasemust be an even function of frequency, for example, a quadraticfunction, a 4^(th) order dependence or other functions with evensymmetry about a center frequency can be utilized. Center frequency isthe zero frequency in the case of conventional images. Equivalently, thederivative of phase profile φ[p,q], called Phase DerivativePD[p,q]=∂φ[p,q]/(∂p·∂q), should be an odd function with respect to p andq. Phase Derivative is also known as group delay in analogy to the sameparameter applied to temporal signals. A linear group delay can be used.Alternatively, one of the simplest (e.g., least number of parameters)yet effective of profiles is the inverse tangent function:

PD[p,q]=a ₁·tan⁻¹(b ₁ ·p)+a ₂·tan⁻¹(b ₂ ·q),  (3)

where tan⁻¹ is the inverse tangent function and a1, b1, a2 and b2 arereal-valued numbers. The total amount of phase φ[p,q] determines thesharpness of the image edges that will be extracted.

The Phase Transformation in the spatial domain can be described asfollows:

$\begin{matrix} & (4)\end{matrix}$

where the phase Kernel in the spatial domain, K[n,m], is described by anonlinear phase operation:

K[n,m]=e ^(j·Φ[n,m]).  (5)

The Phase Kernel K [n, m] in spatial domain is the Fourier transform ofthe Kernel in frequency domain described above. The derivative of phaseprofile in spatial domain Φ[n,m], can be expressed in terms of itsspatial derivative, called Local Frequency LF[n, m]=∂Φ[n,m]/(∂n·∂m). Oneof the simplest (e.g., least number of parameters) yet effective profileis the tangent function:

LF[n,m]=a ₁·tan(b ₁ ·n)+a ₂·tan(b ₂ ·m),  (6)

where tan is the tangent function a1, b1, a2 and b2 are real-valuednumbers. The total amount of phase Φ[n,m] determines the sharpness ofthe image edges that will be extracted. It should be appreciated,however, that other profiles can be utilized without departing from thepresent disclosure.

The Phase Transformation generates the phase image

[n,m] which is further post processed. For edge detection,post-processing in at least one embodiment includes, generating thehistogram of phase image

[n,m], and performing one or more thresholding steps and morphologicaloperations. The histogram shows the density of different edges in theimage on which thresholding is applied. Thresholding is utilized todistinguish between different kinds of edges, for example softer orsharper edges. Morphological operations can be utilized as desired tothin the edges, clean the phase image by removing the isolated pixels,to find prime lines representing edges, or find image corners (points ofinterest). In at least one embodiment, the morphological operations areapplied to the phase output (preferably generated in response topre-filtering, phase-transform, and thresholding), using the techniqueof non-maximum suppression. In at least one embodiment, themorphological operation is configured with hysteresis thresholding forrejecting isolated artifacts that may be otherwise seen at the edges ofthe image. In at least one embodiment, a Scale Invariant FeatureTransform (SIFT) is applied to the output, preferably following themorphological operation. It will be appreciated that variousmorphological operations are known, and can be found in theMorphological Operations toolbox in MATLAB software.

FIG. 1 illustrates an example embodiment 10 of the disclosed PhaseTransform for image processing. The image 12 is passed through a PhaseTransformation 14 which generates an output phase image 16. Theresulting phase image is then further Post Processed 18 to extract thedesired information 20, such as edges and shapes.

FIG. 2 illustrates an example embodiment 30 of a Phase Transformimplemented in the frequency domain. The spatial image 32 is convertedinto frequency domain via Fourier transform 34 and the output 36 is thenmultiplied 38 by the Phase Kernel. Output 40 from the phase kernel isthen passed through an inverse Fourier Transform 42 to obtain thetransform of the image back to a spatial domain. The output of PhaseTransform is the phase 44, that is to say it is the angle of the spatialimage.

FIG. 3 illustrates an example embodiment 50 of a Phase Transformimplemented in the spatial domain. In this implementation, the image 52is convolved with a spatial Phase Kernel 54. The output 56 of the PhaseTransform is the angle of the spatial image.

FIG. 4 illustrates an example embodiment 70 of Post Processing stagesused after the Phase Transform. The phase image 72 is received with ahistogram stage 74 generating a histogram 76 of phase image which isthen separated 78 into ranges of density of different edges in theimage. Next the output 80, after separating into ranges, is passedthrough a thresholding process 82 which is used to distinguish betweendifferent kinds of edges, for example softer or sharper edges, which ingeneral can be a binary thresholding or multi-level thresholdingdepending on the application. Output 84 from thresholding can beoptionally processed utilizing morphological operations 86, such as tothin the edge lines, clean the edge image by removing the isolatedpixels, or to find prime lines representing the edges, prior tooutputting an enhanced phase image 88. A list of common morphologicaloperations that can be used in the Post Processing stage of ourtechnology can be found in Morphological Operations toolbox in MATLABsoftware.

FIG. 5 illustrates an example embodiment 90 of Phase Transform and PostProcessing stages used after Phase Transform stage. This figure depictsthese operations in processing the “Lena” image at each stage. An image92 is received at a phase transform block 94 outputting a transformedimage 96, which is histogram processed 98 to output 100 a histogram withhigh and low sub-ranges. The histogram is then input to a block 102 forseparating the image itself into upper and lower subranges 104, withthese images shown in the figure. After this processing, optionalmorphological operations 106 can be performed to provide the outputimages 108 in the high and low sub-ranges.

The above steps are preferably performed on an image processingapparatus or system, such as having computer processing functionality110, having at least one computer processor (CPU) 112 and at least onememory 114 configured for retaining program data and instructions forexecution on processor 112 for carrying out the method steps depictedthroughout this disclosure. It should be appreciated that only for thesake of simplicity of illustration has processing functionality 110 notbeen shown for each flow diagram in this disclosure. However, one ofordinary skill in the art will recognize that image processing ispreferably carried out by computer processors, which may include withoutlimitation various hardware and graphic accelerators, and optionalspecial purpose digital hardware for speeding up or otherwise enhancingthe image processing operations.

FIG. 6A through FIG. 6E depict various phase kernel and phase derivativerelationships, in particular the normalized phase and the PhaseDerivative of the Kernel in 1D and 2D spatial frequency coordinates. Itshould be noted that these images were originally depicted in color,while shown in this application as black and white for convenience ofreproduction, and not by way of limitation. In FIG. 6A, Phase Derivativeis seen in relation to 1D spatial frequency. In FIG. 6B a Phase profileis seen in relation to 2D spatial frequency, with a cross section of itseen in FIG. 6C. In FIG. 6D is shown 2D phase after application of thespatial frequency mask, with its side view cross section seen in FIG.6E.

FIG. 7A through FIG. 7D depict an image comparison between the disclosededge detection method with that of the the Canny edge detection method.The image under analysis is the “Lena” image whose original is seen inFIG. 7A. Results of edge detection in three cases are shown with FIG. 7Bdepicting Canny edge detection with default threshold parameters, FIG.7C depicting Canny with optimized parameters to emphasize the sharpfeatures, and in FIG. 7D the disclosed edge detection technology isshown.

FIG. 8A through FIG. 8D depicts a comparison of the performance of thedisclosed technology with that of the Canny method for edge and texturedetection. The image under analysis is a portion of the “Lena” imagewith fine texture (the hat), as seen in FIG. 8A. Results of edge andtexture detection in three cases are shown. In FIG. 8B, a Canny outputis shown with optimized parameters to de-emphasize the sharp features,while in FIG. 8C a Canny output is seen with optimized parameters toemphasize the sharp features. In FIG. 8D, edge and texture detection isseen as generated by the disclosed technology.

FIG. 9A through FIG. 9D depicts a performance comparison between thedisclosed technology and that of the Canny method for edge, shape andfeature detection. The image under analysis is a ship upon the ocean asseen in original image of FIG. 9A. Results of edge detection in threecases are shown, Canny with default threshold parameters in FIG. 9B,Canny with optimized parameters to emphasize the sharp features in FIG.9C, and the disclosed technology in FIG. 9D illustrating improved edgedetection and recognition allowing the ship to be easily discerned.

It should be appreciated that images being processed in certainapplications, such as radar generated images, can have complex valuedinput images (data), with the output being another complex-valued datacomprising amplitude and phase, with phase utilized for objectdetection, or tracking, or motion estimation, or edge detection, or anydesired combination thereof. In at least one embodiment, the phasetransform can be implemented in response to operating on the image datawith a mixer and local oscillator having a warped, non-linear, chirp.

FIG. 10A through FIG. 10D depicts a performance comparison between thedisclosed technology and the Canny method when applied to tissueanalysis. The image under analysis is a pathology slide from a lungtissue, as seen in FIG. 10A. Results of edge detection in three casesare shown, with Canny having default settings in FIG. 10B, Canny withoptimized parameters to emphasize the sharp features in FIG. 10C, andthe disclosed edge detection method seen in FIG. 10D as applied here todigital pathology. It can be seen in the image comparison thatstructures that cannot be discerned from the Canny method are readilyapparent in the image generated from the disclosed method.

FIG. 11A through FIG. 11D depict a performance comparison between thedisclosed technology compared to the Canny method when used for brainmapping. The image under analysis is a rat Purkinje neuron injected witha fluorescent dye as seen in FIG. 11A. Results of edge detection inthree cases are shown, Canny with default settings in FIG. 11B, Cannywith optimized parameters to emphasize the sharp features in FIG. 11C,and in FIG. 11D the disclosed edge detection and feature extraction ofthe disclosed technology. The branching structures are more readilyapparent in FIG. 11D than in the images generated with the Canny methodas seen in FIG. 11B and FIG. 11C.

FIG. 12A through FIG. 12D depict a performance comparison between thedisclosed technology and the Canny method when applied to detection inan impaired image. In this example the impairment comprises an uppersection of the image that is partially masked. The objective of the edgeand feature extraction is to detect fine features under the mask. InFIG. 12B an image is seen which was output from a Canny process withdefault settings, while FIG. 12C shows an image output from Canny usingoptimized parameters to emphasize the sharp features. In FIG. 12D anoutput from the disclosed technology is seen for feature, object andtexture detection in an impaired image. Again, it is readily seen thatthe disclosed technology improves detection of fine features in theimage as well as under the mask.

FIG. 13A through FIG. 13C depict a performance comparison between thedisclosed technology and that of Canny when applied to detection in fogas seen in original image 13A. In FIG. 13B image output is seen fromCanny having optimized parameters to emphasize sharp features, which iscompared with the output in FIG. 13D from the disclosed technology. Itwill be noticed that the fog seen in the original image which somewhatobscures the bicyclist near the centerline, is readily discerned fromthe output of the present technology, while this feature is hidden inthe image generated by the Canny method.

FIG. 14 through FIG. 17 depicts screen shots of interactiveedge-detection in numerical microscopy of the present disclosure, suchas performed on a numerical phase-imaging microscope. It should berecognized that the attached images, although shown in black and whitein this application, were in actuality color images, as were a number ofthe images seen throughout this application. In FIG. 14 an originalinput image is seen, while in FIG. 15 “color edge image” has beenselected, whereas the edges are seen, with the selected parameters ofColor Contrast, Sharpness, Phase Warp, Phase Strength, minimumThreshold, and Maximum Threshold. This image output shows edges detectedon all three colors. Edges are shown as analog values that is to saythat no thresholding or morphological operation is performed in thisexample. In this image color contrast is set with R, G and B all at100%. In FIG. 16 and FIG. 17, the same Color Edge Image is seen, butwith only the green (FIG. 16) and red (FIG. 17), with color contrast setat 100%, however, the screenshot shows the same parameters otherwise. Inat least one embodiment, these images are processed by decomposing asource digital image to its constituent colors, followed by optionalpre-filtering for noise reduction, prior to performing a phaseoperation, and any optional thresholding and morphological operations oneach color.

A number of additional operations are described below that areapplicable to images in FIG. 16 and FIG. 17, as well as the otherapplication directed embodiments described. In one preferred embodiment,the user is allowed to adjust the amplitude for each color, as seen inthe lower left corner of FIG. 16 and FIG. 17. In at least one embodimenta minimum intensity trigger is used on the source image to remove lowintensity pixels and improve the signal to noise ratio, as seen by theminimum threshold selection in FIG. 16 and FIG. 17. In at least oneembodiment, a morphological operation is utilized which is configuredfor finding and discarding isolated pixels, and identifying andmaintaining continuous lines and curves in the output image. In at leastone of the embodiments, the disclosed method/apparatus is configured toallow the user to select an image scale (zoom). In this embodiment, theparameters for steps of pre-filtering, Phase Transform, thresholding andmorphological operation are preferably updated automatically based onthis user-selected scale at which the output image is displayed, forimproving identification and display of features in the image.

FIG. 18 through FIG. 21 depict operations of the disclosed edge andfeature extraction as applied to a photograph, or video frame, whoseoriginal image is seen in FIG. 18. In FIG. 19 a “Gray Scale Edge Image”has been selected, while in FIG. 20 a “Color Edge Image” was selected.In FIG. 21 an “Overlay Image” has been selected with Opacity set for theimage and the edge at 100%, whereby the output shows edges of all colorsoverlaid with the original image, and have been processed with boththresholding and morphological operations on the edge image.

FIG. 22 and FIG. 23 illustrate operation on a color image of longexposure busy city street scene seen in original image of FIG. 22. InFIG. 23 the disclosed method of “Color Edge Image” was selected and theoutput image shows edges of all three colors. The edges are shown asanalog values, with no thresholding or morphological operations in thisexample.

FIG. 24 through FIG. 26 illustrate an astronomy example of a celestialbody seen in original image in FIG. 24. In FIG. 25 “Color Edge Image”according to the disclosed technology was selected with the outputshowing edges in all three colors. In this example the edges are shownas analog values, with no thresholding or morphological operationsperformed. In FIG. 26 “Gray Scale Edge Detection” was performedaccording the presented disclosure, whereby the output shows edges of agray-scale image. In this example, the edges are shown as analog valueswith no thresholding or morphological operations utilized.

FIG. 27 and FIG. 28 illustrates a medical example of the disclosedtechnology, shown applied to an original cranial image in FIG. 27. InFIG. 28 a “Gray Scale Edge Image” was selected again with edges shown asanalog values, and without the use of thresholding or morphologicaloperations.

FIG. 29A through FIG. 29D illustrate an example of using the presentedtechnology for performing 3D reconstruction on an original brain imagein FIG. 29A. Output is seen in FIG. 29B as a selected “Overlay” showingedges detected on a brain MRI image (frame number 1) which has beenoverlaid on the source image. In FIG. 29C “Edge” has been selected thatbring outs more edge details that were enhanced by using thresholding ormorphological operations. In FIG. 29D a user interface is seen allowingthe user to control various selections of the edge and featureextraction process, including playing a video sequence, selecting aframe of a video sequence, seeing multiple frames, generating a slideview (gray scale), generating a slice view (contour), generating anoverall structure with edge, generating a cross-section isosurface.Kernel parameters for both S and W can be selected. Video edgeselections can be performed, and pre and post processing actionsselected, which by way of example and not limitation comprise denoisingfactor, sharpness, and minimum and maximum threshold.

FIG. 30A through FIG. 30D illustrate another medical brain applicationof the disclosed edge detection on brain MRI images seen as originalimage in FIG. 30A. An overlay is selected in FIG. 306. In FIG. 30C imageoutput is seen in a 3D output showing edges detected on brain MRI images(horizontal frames number 1 and 26 and vertical frames number 100 and160 in y direction) shown on the cross-sections of isosurface createdfrom brain MRI images. Edges are shown as analog values with Jetcolormap, with no added thresholding or morphological operationsperformed. In the 3D imaging seen in FIG. 30C, the edge detection methodis preferably applied to individual 2D frames, constituting 2D crosssections of a 3D object, which is followed by optional sequentialplayback of the sequence of 2D edge images, and/or reconstruction anddisplay of a 3D edge image from the 2D edge image sequence. It will beappreciated that although exemplified in regard to medical imaging, the3D reconstruction described above can be utilized with any desiredapplications, such as within a numerical phase-imaging microscope,images from which are seen in FIG. 15 through FIG. 17.

2. Edge Detection Using Phase Stretch Transform (PST)

A phase operation is utilized in these embodiments which has somesimilarity to electromagnetic wave propagation through a diffractivemedium. It has been appreciated in the present disclosure that themagnitude of the transformed image's complex amplitude can be used fordata compression when the warp profile follows a specific shape. Inaddition, we have found that the phase of the transform also has uniqueand important properties. In particular, it is demonstrated that thephase can be utilized to create a new and effective edge detectiontechnique. This is achieved by combining the phase properties of theAnamorphic Transform with localization kernel and with morphologicalpost processing. It will be noted that the use of the so-called DiscreteAnamorphic Stretch Transform (OAST) is one form of anamorphic transformemulating propagation of electromagnetic waves through a diffractivemedium with a dielectric function that has warped dispersive (frequencydependent) property.

The present disclosure refers to this edge detection method as a PhaseStretch Transform (PST). PST can be described in both the frequencydomain and the spatial domain as presented below.

2.1. Technical Description

FIG. 31 illustrates an example embodiment 150 for performing edgedetection using a phase stretch transformation. In this method, theoriginal image 152 is first smoothed using a localization kernel(filter) 154, and then is passed through a nonlinear frequency dependentphase operation (phase stretch) 156. The phase of the output compleximage 158 is used for edge detection. The resulting phase profile hasboth positive and negative values. In a post processing stage for edgedetection, the negative values are set to zero in this example. Thehistogram 160 of the phase profile is determined to find the threshold162 for edge detection. After thresholding, the binary image is furtherprocessed by morphological operations 164, to generate an image ofdetected edges 166.

2.1.1. Operation in Frequency Domain

The image under analysis is represented by B└n,m┘ where n and m are twodimensional spatial variables. The PST in the frequency domain can bedescribed as follows:

A[n,m]=

IFFT{{tilde over (K)}[p,q]·{tilde over (L)}[p,q]·FFT2{B[n,m]}}

  (7)

where A[n,m] is the output phase image,

is the angle operator, FFT2 is the two dimensional Fast FourierTransform, (FFT2 is the two dimensional Inverse Fast Fourier Transformand p and q are two dimensional frequency variables. {tilde over(L)}[p,q] is the frequency response of the localization kernel and thewarped phase kernel {tilde over (K)}[p,q] describes a nonlinearfrequency dependent phase:

{tilde over (K)}[p,q]=e ^(j·φ[p,q])  (8)

For edge detection applications the derivative of frequency-dependentphase φ[p,q], called phase derivative PD[p,q] should have a sub-linearfunction with respect to p and q frequency variables. Phase derivativeis also known as group delay in analogy to the same parameter applied totemporal signals. A simplest (e.g., represented by least number ofparameters) profiles is the inverse tangent function:

PD[p,q]=a ₁·tan⁻¹(b ₁ ·p)+a ₂·tan⁻¹(b ₂ ·p),  (9)

where tan⁻¹ ( ) is the inverse tangent function and a₁, b₁, a₂, b₂, arereal-valued numbers. The total amount of phase φ[p,q] and the slope ofthe phase derivative profile at p=q=0 along with the width of thelocalization kernel determine the sharpness of the image edges that willbe extracted.

Application of PST to the image creates the phase image A[n,m] which isfurther post processed. For edge detection, post-processing includesgenerating the histogram of phase image A[n, m], cutting the negativephase values, and optionally performing thresholding and morphologicaloperations. The histogram shows the density of different edges on whichthresholding is applied. Thresholding is used to distinguish betweenvarious edges, e.g., softer or sharper edges. Morphological operationscan be used if needed to thin the edges, clean the phase image byremoving the isolated pixels, or to find prime lines representing edges.

2.1.2. Operation in Spatial Domain

The PST described in Eq. (7) can be also described in the spatial domainas follows:

A[n,m]=

Σ_(j) ₁ _(,j) ₂ _(=−∞) ^(∞)Σ_(k) ₁ _(,k) ₂ _(=−∞) ^(∞) K[n−j ₁ ,m−j₂]·L┌j ₁ −k ₁ ,j ₂ −k ₂ ┐·B┌k ₁ ,k ₂┐

  (10)

where the warped phase kernel in spatial domain, K[n,m] is described bya nonlinear phase operation,

K[n,m]=e ^(j·Φ[n,m])  (11)

and is the Fourier transform of the frequency phase kernel {tilde over(K)}[p,q]. For edge detection applications the derivative of phaseprofile Φ[n,m], called Local Frequency LF[n,m], should have asuper-linear profile with respect to the spatial coordinates n and m. Asimple (e.g., least number of parameters) profile is the tangentfunction:

LF[n,m]=c ₁·tan(d ₁ ·n)+c ₂·tan(d ₂ ·m),  (12)

where tan ( ) is the tangent function and c₁, d₁, c₂ and d₂ arereal-valued numbers.

2.1.3. Designing the Phase Kernel for Edge Detection

The parameters utilized in this embodiment for the disclosed edgedetection methods are:

(1) {a₁, b₁, a₂ and b₂}: parameters of the warped phase kernel;

(2) Δf: bandwidth of the localization kernel; and

(3) Thresh: threshold value.

In FIG. 6A and FIG. 66 as previously described, a typical phasederivative and phase profile for the phase kernel {tilde over (K)}[p,q]is illustrated that results in edge detection. The parameters of theplot are normalized.

The kernel applies a phase that increases with spatial displacement.Since edges contain high frequencies, they are assigned a larger phaseand are then spatially highlighted in the phase of the transformedimage. Parameters of the kernel (a₁b₁ and a₂b₂) control this process.There exists a tradeoff between resolution and SNR. A larger phaseresults in better SNR but at the expense of resolution. Also a higherslope of the phase derivative at the origin results in a sharper edgesbut it also increases the noise. These parameters can be adjustedmanually or optimized iteratively. They can be globally fixed or locallyoptimized.

Frequency bandwidth or spatial length of the localization kernel isdesigned to reduce the noise in the edge detection algorithm but not toremove vital edge information. The threshold value is designed using thehistogram of the phase image after the transform. Dependent on theapplication the threshold can be set to allow more or less edges to beshown in the binary edge image.

2.2. Experimental Results

In this section, examples of detection using the disclosed PST methodare described. For qualitative benchmarking, results are also shown ofedge detection using Canny and Sobel methods. The normalized phasekernel profile and its derivative shape used for the examples presentedin this disclosure were previously shown in FIG. 6A and FIG. 66.

FIG. 32A through FIG. 32D depict a comparison between Canny and thedisclosed PST edge detection method as a first performance example. Theimage under analysis is a gray scale Lena image with 512×512 pixels inTIFF format. In FIG. 32A and FIG. 32B threshold values of 0.09 and 0.15,respectively, for edge detection using the Canny method having a sigmavalue of 0.02. In this case a low sigma value was chosen to enhance thespatial resolution, and two different high threshold values wereselected to illustrate the impact of this parameter (the low thresholdvalues is 25% of the high value by default). For edge detection usingthe disclosed method a phase kernel has been utilized as shown in Eq.(9) with a₁=a₂=5.8182×10⁻⁷ and b₁=b₂=0.6148. When the disclosed methodis combined with a localization kernel with parameter Δf=0.8 and athreshold value of 0.025 utilized the results are shown in FIG. 32C. Itshould be appreciated that morphological operations used to arrive atthe result shown in FIG. 32C include edge thinning and isolated pixelremoving. To analyze the effect of localization kernel and thresholdingvalue, edge detection using Δf=0.4 and threshold value of 0.03 has alsobeen shown in FIG. 32D. As evident in these figures some edges are moreproperly extracted using the disclosed method than when using the Cannyedge detector. Also a comparison of FIG. 32C and FIG. 32D illustratesthat by changing the localization kernel width and threshold values,edges with different strengths can be extracted.

FIG. 33A through FIG. 33D illustrate a comparison between an originalimage of coins and different edge detection methods. The image utilizedfor this comparative analysis is a gray scale image of coins with246×246 pixels in a TIFF format as seen in an original image in FIG.33A. Results of edge detection in three cases are shown, Sobel with athreshold value of 0.042 is seen in FIG. 336, Canny with signal value of0.02 and threshold value of 0.017 is seen in FIG. 33C, and the disclosedmethod is seen in FIG. 33D. For edge detection using the disclosedmethod a phase kernel has been used as shown in Eq. (9) witha₁=a₂=1.3375×10⁻⁹ and b₁=b₂=1.2823, localization kernel parameter Δf=1.6and threshold value of 0.03. The figures confirm that the disclosedmethod has a high resolution and accuracy for finding the edges comparedto Sobel and Canny methods.

Finally the disclosed PST method is analyzed for edge detection onimages taken in low contrast environment. In these situations edgedetection is challenging because the image has low contrast whichreduces the signal to noise ratio of edge detection methods.

FIG. 34A through FIG. 34D illustrate comparison of zooming in edgedetection. In FIG. 34A is an original gray scale image of a rocky beachscene with 512×512 pixels in a TIFF image format. Result of edgedetection using the disclosed method is shown in FIG. 34B, using a phasekernel described in Eq. (9) with a₁=a₂=11.6364 and b₁=b₂=0.6148,localization kernel parameter Δf=20 and threshold value of 0.015. Forbetter illustration, an overlay of the original image has been shownwith the detected edge in two sections of the original image with lowcontrast areas delineated with boxes 1 and 2 in FIG. 34A. Results areshown in FIG. 34C and FIG. 34D. For better comparison, in the overlayimages the original image brightness has been multiplied by two. Asillustrated in this example the disclosed method has found the edgesaccurately in the low contrast part of the image.

The enhancements described in the presented technologies above can bereadily implemented within various image processing systems, whichinclude one or more computer processor devices (e.g., CPU,microprocessor, microcontroller, computer enabled ASIC, etc.) andassociated memory storing instructions (e.g., RAM, DRAM, NVRAM, FLASH,computer readable media, etc.) whereby programming (instructions) storedin the memory are executed on the processor to perform the steps of thevarious process methods described herein.

The computer and memory devices were not depicted in each of thediagrams for the sake of simplicity of illustration, while one ofordinary skill in the art recognizes the use of computer devices forcarrying out steps involved with image/video processing and output. Thepresented technology is non-limiting with regard to memory andcomputer-readable media, insofar as these are non-transitory, and thusnot constituting a transitory electronic signal.

It will also be appreciated that the computer readable media (memorystoring instructions) in these computations systems is “non-transitory”,which comprises any and all forms of computer-readable media, with thesole exception being a transitory, propagating signal. Accordingly, thedisclosed technology may comprise any form of computer-readable media,including those which are random access (e.g., RAM), require periodicrefreshing (e.g., DRAM), those that degrade over time (e.g., EEPROMS,disk media), or that store data for only short periods of time and/oronly in the presence of power, with the only limitation being that theterm “computer readable media” is not applicable to an electronic signalwhich is transitory.

Embodiments of the present technology may be described with reference toflowchart illustrations of methods and systems according to embodimentsof the technology, and/or algorithms, formulae, or other computationaldepictions, which may also be implemented as computer program products.In this regard, each block or step of a flowchart, and combinations ofblocks (and/or steps) in a flowchart, algorithm, formula, orcomputational depiction can be implemented by various means, such ashardware, firmware, and/or software including one or more computerprogram instructions embodied in computer-readable program code logic.As will be appreciated, any such computer program instructions may beloaded onto a computer, including without limitation a general purposecomputer or special purpose computer, or other programmable processingapparatus to produce a machine, such that the computer programinstructions which execute on the computer or other programmableprocessing apparatus create means for implementing the functionsspecified in the block(s) of the flowchart(s).

Accordingly, blocks of the flowcharts, algorithms, formulae, orcomputational depictions support combinations of means for performingthe specified functions, combinations of steps for performing thespecified functions, and computer program instructions, such as embodiedin computer-readable program code logic means, for performing thespecified functions. It will also be understood that each block of theflowchart illustrations, algorithms, formulae, or computationaldepictions and combinations thereof described herein, can be implementedby special purpose hardware-based computer systems which perform thespecified functions or steps, or combinations of special purposehardware and computer-readable program code logic means.

Furthermore, these computer program instructions, such as embodied incomputer-readable program code logic, may also be stored in acomputer-readable memory that can direct a computer or otherprogrammable processing apparatus to function in a particular manner,such that the instructions stored in the computer-readable memoryproduce an article of manufacture including instruction means whichimplement the function specified in the block(s) of the flowchart(s).The computer program instructions may also be loaded onto a computer orother programmable processing apparatus to cause a series of operationalsteps to be performed on the computer or other programmable processingapparatus to produce a computer-implemented process such that theinstructions which execute on the computer or other programmableprocessing apparatus provide steps for implementing the functionsspecified in the block(s) of the flowchart(s), algorithm(s), formula(e),or computational depiction(s).

It will further be appreciated that “programming” as used herein refersto one or more instructions that can be executed by a processor toperform a function as described herein. The programming can be embodiedin software, in firmware, or in a combination of software and firmware.The programming can be stored local to the device in non-transitorymedia, or can be stored remotely such as on a server, or all or aportion of the programming can be stored locally and remotely.Programming stored remotely can be downloaded (pushed) to the device byuser initiation, or automatically based on one or more factors. It willfurther be appreciated that as used herein, that the terms processor,central processing unit (CPU), and computer are used synonymously todenote a device capable of executing the programming and communicationwith input/output interfaces and/or peripheral devices.

From the description herein, it will be appreciated that that thepresent disclosure encompasses multiple embodiments which include, butare not limited to, the following:

1. A method of performing edge detection on a digital image, comprising:(a) receiving a digital image within an image processing device; (b)applying a nonlinear frequency dependent phase operation to the digitalimage; (c) applying a phase kernel during said phase operation so thattotal amount of phase applied in said phase kernel determines sharpnessof image edges being extracted from the digital image; (d) wherein edgedetection is based on phase within a complex image after application ofthe phase kernel from said nonlinear frequency dependent phaseoperation; and (e) generating an image output as a phase image outputwith phase of the transformed image as output.

2. The method of any preceding embodiment, applying said nonlinearfrequency dependent phase operation as a phase transformation, which isapplied in either a frequency domain or a spatial domain, to thereceived digital image.

3. The method of any preceding embodiment, wherein said phasetransformation is applied in the frequency domain comprising performinga transformation of the received digital image to the frequency domainutilizing a phase kernel, and followed by performing an inversetransform to generate a phase image in the spatial domain.

4. The method of any preceding embodiment, further comprising performingsaid phase transformation in the spatial domain by application of alocal frequency, as a phase derivative with respect to spatialcoordinates.

5. The method of any preceding embodiment, wherein said phasetransformation is applied in the spatial domain in response toconvolving with a spatial phase kernel.

6. The method of any preceding embodiment, wherein said phasetransformation is performed utilizing a transform selected from thegroup of transforms consisting of Fourier Transform, Fast FourierTransform (FFT), Discrete Fourier Transform (DFT), Discrete SineTransform (DST) and Discrete Cosine transforms (DCT).

7. The method of any preceding embodiment, further comprising applyingsaid nonlinear frequency dependent phase operation as a phase stretchtransformation, which is applied in either a frequency domain or aspatial domain, to the digital image.

8. The method of any preceding embodiment, further comprising applying alocalization kernel prior to said phase stretch transformation.

9. The method of any preceding embodiment, further comprising settingnegative values in said generated complex phase image to zero.

10. A method of performing edge detection on a digital image,comprising: (a) receiving a digital image within an image processingdevice; (b) applying a nonlinear frequency dependent phase operation asa phase transformation, applied in a frequency domain, to the digitalimage; (c) applying a phase kernel during said phase transformation inwhich total amount of phase applied in said phase kernel determinessharpness of image edges being extracted from the digital image; (d)wherein edge detection is based on phase within a complex image afterapplication of the phase kernel from said nonlinear frequency dependentphase operation; and (e) generating an image output as a phase imagewith phase of the transformed image as output.

11. The method of any preceding embodiment, wherein said phasetransformation is applied in the frequency domain comprising performinga transformation of the input image to the frequency domain utilizing aphase kernel, and followed by performing an inverse transform togenerate a phase image in the spatial domain.

12. A method of performing edge detection on a digital image,comprising: (a) receiving a digital image within an image processingdevice; (b) applying a nonlinear frequency dependent phase operation, asa phase transformation in the spatial domain, to the digital image; (c)applying a phase kernel during said phase operation in response toconvolving with a spatial phase kernel so that total amount of phaseapplied in said phase kernel determines sharpness of image edges beingextracted from the digital image; (d) wherein edge detection is based onphase within a complex image from said nonlinear frequency dependentphase operation; and (e) generating an image output as a phase imagewith phase of the transformed image as output.

13. The method of any preceding embodiment, further comprising postprocessing comprising separating the phase of the transformed image intotwo subranges.

14. The method of any preceding embodiment, further comprising postprocessing comprising separating the phase of the transformed image intomultiple subranges.

15. The method of any preceding embodiment, wherein said subrangescomprise at least one subrange higher than the peak of a phasehistogram, and at least one subrange lower than the peak of a phasehistogram.

16. The method of any preceding embodiment, further comprisingthresholding applied during post processing, wherein said thresholdingfurther selects specific ranges of phase values in creating said phaseimage output.

17. The method of any preceding embodiment, further comprisingthresholding configured for applying at least one threshold level to thephase image.

18. The method of any preceding embodiment, further comprisingmorphological operations performed during post processing to enhancephase image output.

19. The method of any preceding embodiment, further comprising postprocessing of the phase image output by generating a histogram of phaseimage to which thresholding is applied prior to one or moremorphological image operations.

20. The method of any preceding embodiment, further comprisingdistinguishing between different kinds of edges during saidthresholding, including distinguishing between edges that are eithersofter or sharper.

21. The method of any preceding embodiment, further comprisingperforming morphological operations on said phase image output, whereinsaid morphological operations are selected from the group ofmorphological operations consisting of thinning the edges, cleaning aphase image by removing isolated pixels, finding prime linesrepresenting edges, or finding image corners as points of interest, andcombinations thereof.

22. The method of any preceding embodiment, wherein said phase kernel isconfigured with a phase profile that is an even function of frequency.

23. The method of any preceding embodiment, wherein said phase profilecomprises a quadratic profile.

24. The method of any preceding embodiment, further comprising applyingsaid method within a numerical phase-imaging microscope.

25. The method of any preceding embodiment, further comprisingdecomposing a received digital image to its constituent colors prior toperforming said nonlinear frequency dependent phase operation.

26. The method of any preceding embodiment, further comprising applyingsaid frequency dependent phase operation to each of the constituentcolors to generate a phase image output for each color.

27. The method of any preceding embodiment, further comprisingdisplaying generated phase image output for each color eitherseparately, or in combination with one another.

28. The method of any preceding embodiment, further comprisingperforming a pre-filtering step prior to performing said nonlinearfrequency dependent phase operation.

29. The method of any preceding embodiment, further comprising applyingsaid pre-filtering by utilizing at least one linear filter, or at leastone non-linear filter, or a combination of at least one linear andnon-linear filter, wherein said linear and/or non-linear filters areselected as having a desired frequency response.

30. The method of any preceding embodiment, further comprising applyinga minimum intensity trigger on the received digital image to remove lowintensity pixels and to improve signal to noise ratio.

31. The method of any preceding embodiment, further comprising applyinga morphological operation to said phase image output, wherein saidmorphological operation is configured for finding and discardingisolated pixels and identifying and maintaining continuous lines andcurves.

32. The method of any preceding embodiment, further comprisingestablishing a user selected scale for the phase image output, as basedon user input.

33. The method of any preceding embodiment, further comprisingautomatically updating the operations leading up to generation of aphase image output, including nonlinear frequency dependent phaseoperation, based on said user establishing a scale for the phase imageoutput, as based on user input of a user selected scale.

34. The method of any preceding embodiment, wherein said received imagedata comprises either real-valued, or complex-valued, image data.

35. The method of any preceding embodiment, wherein said phase imageoutput comprises complex-valued image data.

36. The method of any preceding embodiment, wherein said complex-valuedimage data includes amplitude and phase.

37. The method of any preceding embodiment, further comprising utilizingsaid phase from said complex-valued image data for performing objectdetection, or tracking, or motion estimation, or edge detection, or anycombination of object detection, tracking, motion estimation, or edgedetection.

38. The method of any preceding embodiment, wherein said nonlinearfrequency dependent phase operation is performed in response tooperating on said received image data with a mixer in combination with alocal oscillator having a warped, non-linear, chirp.

39. The method of any preceding embodiment, further comprisingperforming a morphological operation of non-maximum suppression on saidphase image output.

40. The method of any preceding embodiment, further comprisingperforming a morphological operation of Hysteresis thresholding on saidphase image output for rejecting isolated artifacts to prevent them frombeing recognized as edges.

41. The method of any preceding embodiment, further comprisingperforming a post-processing operation of Scale Invariant FeatureTransform (SIFT).

42. The method of any preceding embodiment, wherein said method ofperforming edge detection on a digital image is configured forgenerating two-dimensional (2D), or three-dimensional (3D) phase imageoutput.

43. The method of any preceding embodiment, wherein three-dimensional(3D) phase image output, comprises applying edge detection to individual2D frames of received image data, which are cross-sections of a 3Dobject, and then reconstructing these 2D cross-sections into a 3D phaseimage output.

44. The method of any preceding embodiment, wherein said method ofperforming edge detection on a digital image is configured for beingutilized for image processing selected from the group of imageprocessing functions consisting of: (a) image enhancement, (b) edge andcorner detection, (c) object, shape, pattern and texture detection andrecognition, (d) tracking and motion estimation, (e) 2D or 3D still orvideo image processing, (f) real-time sensing, (g) visual accommodation,(h) rendering and perception, (i) computer vision and machine learning,(j) tissue diagnostics in digital pathology and radiology, and (k)imaging through fog and other diffusive media.

45. An apparatus for performing edge detection on a digital image,comprising: (a) at least one processor; (b) memory storing instructions;(c) said instructions when executed by the processor performing stepscomprising: (c)(i) receiving a digital image for image processing;(c)(ii) applying a nonlinear frequency dependent phase operation to thedigital image; and (c)(iii) applying a phase kernel during said phaseoperation so that total amount of phase applied in said phase kerneldetermines sharpness of image edges being extracted from the digitalimage; (c)(iv) wherein edge detection is based on phase within a compleximage from said nonlinear frequency dependent phase operation; and(c)(v) generating an image output as a phase image with phase of thetransformed image as output.

46. The apparatus of any preceding embodiment, wherein said nonlinearfrequency dependent phase operation comprises a phase transformationapplied in either a frequency domain or a spatial domain, to the digitalimage.

47. The apparatus of any preceding embodiment, wherein said phasetransformation is applied in a frequency domain comprising instructionswhen executed by the processor for performing a transformation of theinput image to the frequency domain utilizing a phase kernel, andfollowed by performing an inverse transform to generate a phase image inthe spatial domain.

48. The apparatus of any preceding embodiment, wherein said phasetransformation is applied in the spatial domain by instructions thatwhen executed by the processor apply a local frequency as a phasederivative with respect to spatial coordinates.

49. The apparatus of any preceding embodiment, wherein said instructionswhen executed by the processor are configured for applying said phasetransformation in the spatial domain in response to convolving with aspatial phase kernel.

50. The apparatus of any preceding embodiment, wherein said instructionswhen executed by the processor perform said phase transformationutilizing a transform selected from the group of transforms consistingof Fourier Transform, Fast Fourier Transform (FFT), Discrete FourierTransform (OFT), Discrete Sine Transform (DST) and Discrete Cosinetransforms (DCT).

51. The apparatus of any preceding embodiment, wherein said instructionswhen executed by the processor are configured for performing saidnonlinear frequency dependent phase operation as a phase stretchtransformation, which is applied in either a frequency domain or aspatial domain, to the received digital image.

52. The apparatus of any preceding embodiment, further comprisinginstructions that when executed by the processor are configured forapplying a localization kernel prior to said phase stretchtransformation.

53. The apparatus of any preceding embodiment, further comprisinginstructions that when executed by the processor are configured forsetting negative values in said generated complex phase image to zero.

54. An image processing method, comprising: (a) applying a phasetransformation to an input image; and (b) using the phase of thetransformed image as the output.

55. The method of any preceding embodiment, wherein the method is usedfor image enhancement, or for edge and corner detection, or for objectdetection and tracking, or for machine learning, or for artistic specialeffects, or any combination of the foregoing.

56. A method for processing an image, comprising: (a) applying a phasetransform to the image spectrum in the spectral domain; and (b) usingthe phase of the image in the spatial domain as the output.

57. The method of any preceding embodiment, wherein said phase transformin the spectral domain is described by application of a phasederivative, defined as phase derivative with respect to spatialfrequency, that is a nonlinear function of frequency.

58. The method of any preceding embodiment, wherein said phase transformis described by application of a phase derivative, defined as phasederivative with respect to spatial frequency, that is a sublinearfunction of frequency.

59. The method of any preceding embodiment, wherein said phase transformis described by application of a phase derivative, defined as phasederivative with respect to to spatial frequency, that is a sublinearfunction of frequency.

Although the description herein contains many details, these should notbe construed as limiting the scope of the disclosure but as merelyproviding illustrations of some of the presently preferred embodiments.Therefore, it will be appreciated that the scope of the disclosure fullyencompasses other embodiments which may become obvious to those skilledin the art.

In the claims, reference to an element in the singular is not intendedto mean “one and only one” unless explicitly so stated, but rather “oneor more.” All structural and functional equivalents to the elements ofthe disclosed embodiments that are known to those of ordinary skill inthe art are expressly incorporated herein by reference and are intendedto be encompassed by the present claims. Furthermore, no element,component, or method step in the present disclosure is intended to bededicated to the public regardless of whether the element, component, ormethod step is explicitly recited in the claims. No claim element hereinis to be construed as a “means plus function” element unless the elementis expressly recited using the phrase “means for”. No claim elementherein is to be construed as a “step plus function” element unless theelement is expressly recited using the phrase “step for”.

What is claimed is:
 1. A method of performing edge detection on adigital image, comprising: receiving a digital image within an imageprocessing device; applying a nonlinear frequency dependent phaseoperation to the digital image; applying a phase kernel during saidphase operation so that total amount of phase applied in said phasekernel determines sharpness of image edges being extracted from thedigital image; wherein edge detection is based on phase within a compleximage after application of the phase kernel from said nonlinearfrequency dependent phase operation; and generating an image output as aphase image output with phase of the transformed image as output.